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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-clel3gALT | Structured version Visualization version GIF version |
Description: Alternate proof of clel3g 3558. (Contributed by BJ, 1-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-clel3gALT | ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2814 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐵) | |
2 | 1 | biantrurd 536 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ (∃𝑥 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵))) |
3 | 19.41v 1957 | . . 3 ⊢ (∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵)) | |
4 | 2, 3 | bitr4di 292 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵))) |
5 | eleq2 2821 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵)) | |
6 | 5 | bicomd 226 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝑥)) |
7 | 6 | pm5.32i 578 | . . 3 ⊢ ((𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵) ↔ (𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)) |
8 | 7 | exbii 1854 | . 2 ⊢ (∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵) ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)) |
9 | 4, 8 | bitrdi 290 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 |
This theorem is referenced by: (None) |
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